The present application relates to predicting stiffness coefficients (Cij) in a transversely isotropic medium such as laminated shale or shale with orthogonal set of transverse natural fractures.
Hydrocarbon shale formations have very low permeability and, therefore, are hydraulically fractured (i.e., fracked) to enhanced hydrocarbon recovery therefrom. In some instances, the fracturing operations may be simulated to identify target zones in the formation for drilling wells and identifying perforating and fracturing locations (“sweet spots”). Simulating a fracturing operation often involves two steps: a stage/perforation design that models perforations along the wellbore and then a hydraulic fracture model that investigates how fractures extend from perforations in the wellbore into the surrounding formation. The stage/perforation designs typically implemented produce a profile (e.g., as a graph or map) of closure stress versus wellbore depth taking into account rock brittleness, both of which are predicted from the elastic moduli profiles of the formation. The hydraulic fracture modeling also requires the depth profile of closure stress along with the Young's modulus and Poisson's ratio in an input.
Within shale formations, fracturing operation simulations are influenced by extensive horizontal laminations of the formation. The laminations strongly influence the fracture height because of the differences in the rock mechanical properties normal (perpendicular) and parallel to the bedding planes. In order to accurately predict fracturing height from logs in this environment, these mechanical property differences must be taken into account. The mechanical properties normal and parallel to the bedding may be predicted based on logs of the Young's Modulus and Poisson's Ratio parallel to bedding planes (referred to as the horizontal Young's Modulus (Ehorz) and Poisson's Ratio (vhorz), respectively) and also the Young's Modulus and Poisson's Ratio normal to the bedding planes (referred to as the vertical Young's Modulus (Evert) and Poisson's Ratio (vvert), respectively).
The stiffness matrix for anisotropic materials with orthotropic symmetry (e.g., a vertically transverse isotropic (VTI) rock such as laminated shale without natural fractures) may be expressed with various stiffness coefficients (Cij) as Equation (1).
                              C          ijkl                =                  [                                                                      C                  11                                                                              C                  12                                                                              C                  13                                                            0                                            0                                            0                                                                                      C                  12                                                                              C                  22                                                                              C                  23                                                            0                                            0                                            0                                                                                      C                  13                                                                              C                  23                                                                              C                  33                                                            0                                            0                                            0                                                                    0                                            0                                            0                                                              C                  44                                                            0                                            0                                                                    0                                            0                                            0                                            0                                                              C                  55                                                            0                                                                    0                                            0                                            0                                            0                                            0                                                              C                  66                                                              ]                                    Equation        ⁢                                  ⁢                  (          1          )                    
The five independent elastic coefficients of the transversely isotropic media are C11, C33, C12, C13, and C44. Alternatively, the elastic coefficients selected may be C33, C11, C13, C44, and C66, because of the relationship between C66, C11, and C12 (C66=(C11−C12)/2), which is due to the VTI symmetry.
The elastic moduli (referred to as “dynamic moduli” when obtained from velocity measurements) can be determined in both the vertical and horizontal directions using Equations (2)-(5), which allows for anisotropy to be quantified by wireline acoustical logging measurements.
                              E          vert                =                              C            33                    -                                    2              ⁢                              C                13                2                                                                    C                11                            -                              C                12                                                                        Equation        ⁢                                  ⁢                  (          2          )                                                  E          horz                =                                            (                                                C                  11                                -                                  C                  12                                            )                        *                          (                                                                    C                    11                                    ⁢                                      C                    33                                                  -                                  2                  ⁢                                      C                    13                    2                                                  +                                                      C                    12                                    ⁢                                      C                    33                                                              )                                                                          C                11                            ⁢                              C                33                                      -                          C              13              2                                                          Equation        ⁢                                  ⁢                  (          3          )                                                  v          vert                =                              C            13                                              C              11                        +                          C              12                                                          Equation        ⁢                                  ⁢                  (          4          )                                                  v          horz                =                                                            C                33                            ⁢                              C                12                                      +                          C              13              2                                                                          C                33                            ⁢                              C                11                                      -                          C              13              2                                                          Equation        ⁢                                  ⁢                  (          5          )                    
According to Equations (2)-(5), fully characterizing geomechanical properties of laminated shale formations requires deriving or measuring the five independent elastic coefficients. In a vertical well, C33 and C44 are calculated directly from the velocity of the vertically propagating p- and s-waves; C66 is estimated from the Stoneley wave velocity measured by an advanced sonic tool in open hole; and C11 and C13 may be obtained from an empirical model. For example, ANNIE model and modified ANNIE model workflows are provided in FIGS. 1A and 1B, respectively. (ANNIE model was developed in 1996 by Shoenberg, Muir, and Sayers, and is generally understood in the art apart from the inventive aspects disclosed herein).
However, empirical models for characterizing the VTI stiffness tensor require Stoneley wave velocity as input, which prevents their applicability in the cased-hole conditions or when sonic tools without Stoneley wave measurement capacity are used. (A Stoneley wave is generally understood in the art apart from the inventive aspects disclosed herein, and was named after Dr. Robert Stoneley who discovered it.)
Some have proposed a method to do the interpretation without Stoneley wave velocity based on the widely observed near-linear relationships between measured 0°, 45°, and 90° p- and s-wave velocities from different shales (FIGS. 2A and 2B). Based on those linear relationships, wave velocities at different angles can be derived from the log measured wave velocity (e.g., 0° in a vertical well or 90° in a horizontal well). These velocities may then be used with a density log to compute the stiffness coefficients. More specifically, C33 and C44 are directly calculated from the measured p-wave velocity at 0° (Vp(0°)) and the measured s-wave velocity at 0° (Vs(0°)), respectively. C11 and C66 are directly calculated from the predicted p-wave velocity at 90° (Vp(90°)) from FIG. 2A and the predicted s-wave velocity at 90° (Vs(90°)) from FIG. 2B, respectively. C13 is calculated from predicted p-wave velocity at 45° (Vp(45°)) from FIG. 2A. From FIGS. 2A and 2B, it is noted that the 90° velocity has poorer linear relationship with 0° velocity as compared with 45° data. The scatter indicator R2 is around 20% lower. The root-mean-square deviation (RMSD) of 90° velocity is almost double of the RMSD of 45° velocity. These poor linear relationships may lead to significant error when used in stiffness coefficient calculations, which may result in erroneous calculation of stress and elastic moduli, and hence result in inefficient or ineffective fracturing operations.